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vuepress-plugin-mathjax <a rel="noopener noreferrer" target="_blank" title="vuepress/vuepress-plugin-mathjax" href="https://github.com/vuepress/vuepress-plugin-mathjax" class="github-link"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 28 28"><path d="M14 0C6.27 0 0 6.43 0 14.36c0 6.34 4.01 11.72 9.57 13.62.7.13.96-.31.96-.69 0-.34-.01-1.24-.02-2.44-3.89.87-4.72-1.92-4.72-1.92-.64-1.66-1.55-2.1-1.55-2.1-1.27-.89.1-.87.1-.87 1.4.1 2.14 1.48 2.14 1.48 1.25 2.19 3.28 1.56 4.07 1.19.13-.93.49-1.56.89-1.92-3.11-.36-6.38-1.59-6.38-7.09 0-1.57.55-2.85 1.44-3.85-.14-.36-.62-1.82.14-3.8 0 0 1.18-.39 3.85 1.47a12.8 12.8 0 0 1 3.5-.48c1.19.01 2.39.16 3.5.48 2.67-1.86 3.85-1.47 3.85-1.47.76 1.98.28 3.44.14 3.8.9 1 1.44 2.28 1.44 3.85 0 5.51-3.27 6.73-6.39 7.08.5.44.95 1.32.95 2.66 0 1.92-.02 3.47-.02 3.94 0 .38.25.83.96.69C23.99 26.07 28 20.7 28 14.36 28 6.43 21.73 0 14 0z" fill-rule="evenodd" clip-rule="evenodd"></path></svg></a></h1> <p>Support TeX syntax in markdown files of your VuePress site.</p> <h2 id="installation"><a href="#installation" class="header-anchor">#</a> Installation</h2> <div class="language-sh extra-class"><pre class="language-sh"><code><span class="token function">npm</span> <span class="token function">install</span> -D vuepress-plugin-mathjax
</code></pre></div><h2 id="usage"><a href="#usage" class="header-anchor">#</a> Usage</h2> <div class="language-js extra-class"><pre class="language-js"><code><span class="token comment">// .vuepress/config.js</span>
module<span class="token punctuation">.</span>exports <span class="token operator">=</span> <span class="token punctuation">{</span>
  plugins<span class="token operator">:</span> <span class="token punctuation">[</span>
    <span class="token punctuation">[</span>
      <span class="token string">'vuepress-plugin-mathjax'</span><span class="token punctuation">,</span>
      <span class="token punctuation">{</span>
        target<span class="token operator">:</span> <span class="token string">'svg'</span><span class="token punctuation">,</span>
        macros<span class="token operator">:</span> <span class="token punctuation">{</span>
          <span class="token string">'*'</span><span class="token operator">:</span> <span class="token string">'\\times'</span><span class="token punctuation">,</span>
        <span class="token punctuation">}</span><span class="token punctuation">,</span>
      <span class="token punctuation">}</span><span class="token punctuation">,</span>
    <span class="token punctuation">]</span><span class="token punctuation">,</span>
  <span class="token punctuation">]</span><span class="token punctuation">,</span>
<span class="token punctuation">}</span>
</code></pre></div><h2 id="configs"><a href="#configs" class="header-anchor">#</a> Configs</h2> <p>Configurations marked with <span class="badge tip" style="vertical-align:;" data-v-15b7b770>frontmatter</span> will be allowed to be written in the <code>mathjax</code> option of the frontmatter of the page in VuePress 1.0.0-alpha.39 and higher. E.g:</p> <div class="language-yaml extra-class"><pre class="language-yaml"><code><span class="token punctuation">---</span>
<span class="token key atrule">mathjax</span><span class="token punctuation">:</span>
  <span class="token key atrule">presets</span><span class="token punctuation">:</span> <span class="token string">'\def\lr#1#2#3{\left#1#2\right#3}'</span>
<span class="token punctuation">---</span>
</code></pre></div><h3 id="target"><a href="#target" class="header-anchor">#</a> target</h3> <ul><li><strong>type</strong>: <code>'svg' | 'chtml'</code></li> <li><strong>default</strong>: <code>'chtml'</code></li></ul> <p>The output of MathJax.</p> <h3 id="packages"><a href="#packages" class="header-anchor">#</a> packages</h3> <ul><li><strong>type</strong>: <code>string | string[]</code></li> <li><strong>default</strong>: all the MathJax packages available</li></ul> <p>The MathJax packages to use.</p> <h3 id="macros"><a href="#macros" class="header-anchor">#</a> macros</h3> <ul><li><strong>type</strong>: <code>{ [key: string]: string | null }</code></li> <li><strong>default</strong>: <code>{}</code></li></ul> <p>Macros will be automatically mixed with built-in macros. To disable a built-in macro, simply set the value to <code>null</code> accordingly. Here is a list of all built-in macros:</p> <h3 id="presets"><a href="#presets" class="header-anchor">#</a> presets <span class="badge tip" style="vertical-align:top;" data-v-15b7b770>frontmatter</span></h3> <ul><li><strong>type</strong>: <code>string | string[]</code></li> <li><strong>default</strong>: <code>[]</code></li></ul> <p>The preset content to be added. The preset content will automatically be inserted before the TeX code.</p> <h3 id="showerror"><a href="#showerror" class="header-anchor">#</a> showError <span class="badge tip" style="vertical-align:top;" data-v-15b7b770>vuepress 1.0.0-alpha.40+</span></h3> <ul><li><strong>type</strong>: <code>boolean</code></li> <li><strong>default</strong>: <code>process.env.NODE_ENV === 'development'</code></li></ul> <p>Whether to output an error message in the console when a compilation error is encountered.</p> <h3 id="cache"><a href="#cache" class="header-anchor">#</a> cache</h3> <ul><li><strong>type</strong>: <code>false | object</code></li> <li><strong>default</strong>: <code>{}</code></li></ul> <p><a href="https://github.com/isaacs/node-lru-cache" target="_blank" rel="noopener noreferrer">LRU Cache<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a> Options. If set to <code>false</code>, no cache will be used.</p> <h2 id="syntax"><a href="#syntax" class="header-anchor">#</a> Syntax</h2> <h3 id="inline"><a href="#inline" class="header-anchor">#</a> Inline</h3> <p>Surround your LaTeX with a single <code>$</code> on each side for inline rendering.</p> <p><strong>Input:</strong></p> <pre class="math-block">Euler's identity $e^{i\pi}+1=0$ is a beautiful formula in $\mathbb{R}^2$.
</pre> <p><strong>Output:</strong></p> <div class="math-block"><p>Euler's identity <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-msup><mjx-mi class="mjx-i"><mjx-c c="e"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c c="3C0"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo space="3" class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mn space="3" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn><mjx-mo space="4" class="mjx-n"><mjx-c c="="></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="0"></mjx-c></mjx-mn></mjx-math></mjx-container> is a beautiful formula in <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-msup><mjx-TeXAtom><mjx-mi class="mjx-ds"><mjx-c c="R"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-script style="vertical-align:0.41em;"><mjx-mn size="s" class="mjx-n"><mjx-c c="2"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math></mjx-container>.</p></div> <h3 id="block"><a href="#block" class="header-anchor">#</a> Block</h3> <p>Use two ($$) for block rendering. This mode uses bigger symbols and centers the result.</p> <p><strong>Input:</strong></p> <pre class="math-block">$$\frac {\partial^r} {\partial \omega^r} \left(\frac {y^{\omega}} {\omega}\right) 
= \left(\frac {y^{\omega}} {\omega}\right) \left\{(\log y)^r + \sum_{i=1}^r \frac {(-1)^i r \cdots (r-i+1) (\log y)^{r-i}} {\omega^i} \right\}$$
</pre> <p><strong>Output:</strong></p> <div class="math-block"><p></p><p><mjx-container jax="CHTML" display="true" class="MathJax"><mjx-math display="true" class=" MJX-TEX"><mjx-mo class="mjx-n"><mjx-c c="="></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c c="("></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s3"><mjx-c c=")"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow><mjx-mo class="mjx-s4"><mjx-c c="{"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-mi size="s" class="mjx-i"><mjx-c c="r"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo space="3" class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-munderover space="3"><mjx-over style="padding-bottom:0.192em;padding-left:0.563em;"><mjx-mi size="s" class="mjx-i"><mjx-c c="r"></mjx-c></mjx-mi></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c c="2211"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.148em;"><mjx-TeXAtom size="s"><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="="></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi size="s" class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mi class="mjx-i"><mjx-c c="r"></mjx-c></mjx-mi><mjx-mo space="2" class="mjx-n"><mjx-c c="22EF"></mjx-c></mjx-mo><mjx-mo space="2" class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="r"></mjx-c></mjx-mi><mjx-mo space="3" class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mi space="3" class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi><mjx-mo space="3" class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mn space="3" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mi class="mjx-i"><mjx-c c="r"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi size="s" class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s4"><mjx-c c="}"></mjx-c></mjx-mo></mjx-mrow></mjx-math></mjx-container></p><p></p></div> <h3 id="conventions"><a href="#conventions" class="header-anchor">#</a> Conventions</h3> <p>Math parsing in markdown is designed to agree with the conventions set by <a href="http://pandoc.org/MANUAL.html#math" target="_blank" rel="noopener noreferrer">pandoc<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a>:</p> <blockquote><p>Anything between two $ characters will be treated as TeX math. The opening $ must have a non-space character immediately to its right, while the closing $ must have a non-space character immediately to its left, and must not be followed immediately by a digit. Thus, $20,000 and $30,000 won’t parse as math. If for some reason you need to enclose text in literal $ characters, backslash-escape them and they won’t be treated as math delimiters.</p></blockquote> <h2 id="features"><a href="#features" class="header-anchor">#</a> Features</h2> <h3 id="using-macros"><a href="#using-macros" class="header-anchor">#</a> Using Macros</h3> <p>This is part of <code>config.js</code> of this project:</p> <div class="language-js extra-class"><pre class="language-js"><code>module<span class="token punctuation">.</span>exports <span class="token operator">=</span> <span class="token punctuation">{</span>
  plugins<span class="token operator">:</span> <span class="token punctuation">[</span>
    <span class="token punctuation">[</span>
      <span class="token string">'vuepress-plugin-mathjax'</span><span class="token punctuation">,</span>
      <span class="token punctuation">{</span>
        macros<span class="token operator">:</span> <span class="token punctuation">{</span>
          <span class="token string">'\\Z'</span><span class="token operator">:</span> <span class="token string">'\\mathbb{Z}'</span><span class="token punctuation">,</span>
        <span class="token punctuation">}</span><span class="token punctuation">,</span>
      <span class="token punctuation">}</span><span class="token punctuation">,</span>
    <span class="token punctuation">]</span><span class="token punctuation">,</span>
  <span class="token punctuation">]</span><span class="token punctuation">,</span>
<span class="token punctuation">}</span>
</code></pre></div><p><strong>Input:</strong></p> <pre class="math-block">We have $a&gt;n &lt;=&gt; a&gt;=n+1$, if $a, n\in\Z$.
</pre> <p><strong>Output:</strong></p> <div class="math-block"><p>We have <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-i"><mjx-c c="a"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="&gt;"></mjx-c></mjx-mo><mjx-mi space="4" class="mjx-i"><mjx-c c="n"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="21D4"></mjx-c></mjx-mo><mjx-mi space="4" class="mjx-i"><mjx-c c="a"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7E"></mjx-c></mjx-mo><mjx-mi space="4" class="mjx-i"><mjx-c c="n"></mjx-c></mjx-mi><mjx-mo space="3" class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mn space="3" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-math></mjx-container>, if <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-i"><mjx-c c="a"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=","></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="n"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2208"></mjx-c></mjx-mo><mjx-TeXAtom space="4"><mjx-mi class="mjx-ds"><mjx-c c="Z"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math></mjx-container>.</p></div> <h3 id="using-presets"><a href="#using-presets" class="header-anchor">#</a> Using Presets <span class="badge tip" style="vertical-align:top;" data-v-15b7b770>vuepress 1.0.0-alpha.39+</span></h3> <p>This is the frontmatter of this page:</p> <div class="language-yaml extra-class"><pre class="language-yaml"><code><span class="token punctuation">---</span>
<span class="token key atrule">sidebarDepth</span><span class="token punctuation">:</span> <span class="token number">3</span>
<span class="token key atrule">mathjax</span><span class="token punctuation">:</span>
  <span class="token key atrule">presets</span><span class="token punctuation">:</span> <span class="token string">'\def\lr#1#2#3{\left#1#2\right#3}'</span>
<span class="token punctuation">---</span>
</code></pre></div><p><strong>Input:</strong></p> <pre class="math-block">$$
</pre> <p><strong>Output:</strong></p> <div class="math-block"><p></p><p><mjx-container jax="CHTML" display="true" class="MathJax"><mjx-math display="true" class=" MJX-TEX"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mi class="mjx-n"><mjx-c c="2202"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-n"><mjx-c c="2202"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c c="t"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c c="("></mjx-c></mjx-mo><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi size="s" class="mjx-i"><mjx-c c="t"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mi class="mjx-i"><mjx-c c="t"></mjx-c></mjx-mi></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-s3"><mjx-c c=")"></mjx-c></mjx-mo></mjx-mrow></mjx-math></mjx-container></p><p></p></div> <h2 id="miscellaneous"><a href="#miscellaneous" class="header-anchor">#</a> Miscellaneous</h2> <h3 id="dependencies"><a href="#dependencies" class="header-anchor">#</a> Dependencies</h3> <p>This plugin uses <a href="https://github.com/mathjax/mathjax-v3" target="_blank" rel="noopener noreferrer">mathjax-v3<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a> (Early beta) which is not ready for production.</p> <h3 id="related-libraries"><a href="#related-libraries" class="header-anchor">#</a> Related Libraries</h3> <p>This plugin is inspired by some other libraries, thank you!</p> <ul><li><a href="https://github.com/zlliang/vuepress-plugin-latex" target="_blank" rel="noopener noreferrer">vuepress-plugin-latex<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li><a href="https://github.com/waylonflinn/markdown-it-katex" target="_blank" rel="noopener noreferrer">markdown-it-katex<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li><a href="https://github.com/goessner/markdown-it-texmath" target="_blank" rel="noopener noreferrer">markdown-it-texmath<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li><a href="https://github.com/classeur/markdown-it-mathjax" target="_blank" rel="noopener noreferrer">markdown-it-mathjax<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li> <li><a href="https://github.com/yamavol/markdown-it-mathjax-chtml" target="_blank" rel="noopener noreferrer">markdown-it-mathjax-chtml<svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></a></li></ul> <h2 id="demo"><a href="#demo" class="header-anchor">#</a> Demo</h2> <p><strong>Input:</strong></p> <pre class="math-block">Supposing that $y &gt;= 0$ and that $[\log x]$ represents the integer part of $\log x$, let:
$$\Phi (y) = \frac {1} {2 \pi i} \int_{2 - i \infty}^{2 + i \infty} \frac {y^{\omega} \mathrm{d} \omega} {\omega \left(1 + \frac {\omega} {(\log x)^{1.1}}\right)^{[ \log x ] + 1}}, x &gt; 1$$
Obviously, when $0 &lt;= y &lt;= 1$, there is $\Phi(y) = 0$. For all $y &gt;= 0$, $\Phi(y)$ is a non-decreasing function.
When $\log x&gt;=10^4$ and $y&gt;= e^{2{(\log x)}^{-0.1}}$, thus:
$$1 - x^{- 0.1} &lt;= \Phi (y) &lt;= 1$$
</pre> <p><strong>Output:</strong></p> <div class="math-block"><p>Supposing that <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7E"></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="0"></mjx-c></mjx-mn></mjx-math></mjx-container> and that <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mo class="mjx-n"><mjx-c c="["></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="]"></mjx-c></mjx-mo></mjx-math></mjx-container> represents the integer part of <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi></mjx-math></mjx-container>, let:</p> <p></p><p><mjx-container jax="CHTML" display="true" class="MathJax"><mjx-math display="true" class=" MJX-TEX"><mjx-mi class="mjx-n"><mjx-c c="3A6"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-mo space="4" class="mjx-n"><mjx-c c="="></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c c="2"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c c="3C0"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-msubsup space="2"><mjx-mo noIC="true" class="mjx-lop"><mjx-c c="222B"></mjx-c></mjx-mo><mjx-script style="vertical-align:-0.896em;"><mjx-TeXAtom size="s" style="margin-left:0.729em;"><mjx-mn class="mjx-n"><mjx-c c="2"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi><mjx-mi class="mjx-n"><mjx-c c="221E"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-spacer style="margin-top:1.455em;"></mjx-spacer><mjx-TeXAtom size="s"><mjx-mn class="mjx-n"><mjx-c c="2"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="i"></mjx-c></mjx-mi><mjx-mi class="mjx-n"><mjx-c c="221E"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msubsup><mjx-mfrac space="2"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-TeXAtom><mjx-mi class="mjx-n"><mjx-c c="d"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mi class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi><mjx-msup><mjx-mrow><mjx-mo class="mjx-lop"><mjx-c c="("></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn><mjx-mo space="3" class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mfrac space="3"><mjx-frac><mjx-num><mjx-nstrut></mjx-nstrut><mjx-mi size="s" class="mjx-i"><mjx-c c="3C9"></mjx-c></mjx-mi></mjx-num><mjx-dbox><mjx-dtable><mjx-line></mjx-line><mjx-row><mjx-den><mjx-dstrut></mjx-dstrut><mjx-mrow size="s"><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c><mjx-c c="."></mjx-c><mjx-c c="1"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-lop"><mjx-c c=")"></mjx-c></mjx-mo></mjx-mrow><mjx-script style="vertical-align:0.877em;"><mjx-TeXAtom size="s"><mjx-mo class="mjx-n"><mjx-c c="["></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="]"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c c="+"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n"><mjx-c c=","></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="&gt;"></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-math></mjx-container></p><p></p> <p>Obviously, when <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mn class="mjx-n"><mjx-c c="0"></mjx-c></mjx-mn><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7D"></mjx-c></mjx-mo><mjx-mi space="4" class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7D"></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-math></mjx-container>, there is <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-n"><mjx-c c="3A6"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-mo space="4" class="mjx-n"><mjx-c c="="></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="0"></mjx-c></mjx-mn></mjx-math></mjx-container>. For all <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7E"></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="0"></mjx-c></mjx-mn></mjx-math></mjx-container>, <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-n"><mjx-c c="3A6"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo></mjx-math></mjx-container> is a non-decreasing function.</p> <p>When <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7E"></mjx-c></mjx-mo><mjx-msup><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c><mjx-c c="0"></mjx-c></mjx-mn><mjx-script style="vertical-align:0.393em;"><mjx-mn size="s" class="mjx-n"><mjx-c c="4"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-math></mjx-container> and <mjx-container jax="CHTML" class="MathJax"><mjx-math class=" MJX-TEX"><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7E"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c c="e"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s"><mjx-mn class="mjx-n"><mjx-c c="2"></mjx-c></mjx-mn><mjx-msup><mjx-TeXAtom><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-n"><mjx-c c="l"></mjx-c><mjx-c c="o"></mjx-c><mjx-c c="g"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="2061"></mjx-c></mjx-mo><mjx-mi space="2" class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-script style="vertical-align:0.477em;"><mjx-TeXAtom size="s"><mjx-mo class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="0"></mjx-c><mjx-c c="."></mjx-c><mjx-c c="1"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math></mjx-container>, thus:</p> <p></p><p><mjx-container jax="CHTML" display="true" class="MathJax"><mjx-math display="true" class=" MJX-TEX"><mjx-mn class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn><mjx-mo space="3" class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-msup space="3"><mjx-mi class="mjx-i"><mjx-c c="x"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s"><mjx-mo class="mjx-n"><mjx-c c="2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c c="0"></mjx-c><mjx-c c="."></mjx-c><mjx-c c="1"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7D"></mjx-c></mjx-mo><mjx-mi space="4" class="mjx-n"><mjx-c c="3A6"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c="("></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c c="y"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c c=")"></mjx-c></mjx-mo><mjx-mo space="4" class="mjx-n"><mjx-c c="2A7D"></mjx-c></mjx-mo><mjx-mn space="4" class="mjx-n"><mjx-c c="1"></mjx-c></mjx-mn></mjx-math></mjx-container></p><p></p></div></div> <footer class="page-edit"><div class="edit-link"><a href="https://github.com/vuepress/vuepress-community/edit/master/packages/docs/src/en/plugins/mathjax.md" target="_blank" rel="noopener noreferrer">Edit this page on GitHub</a> <svg xmlns="http://www.w3.org/2000/svg" aria-hidden="true" x="0px" y="0px" viewBox="0 0 100 100" width="15" height="15" class="icon outbound"><path fill="currentColor" d="M18.8,85.1h56l0,0c2.2,0,4-1.8,4-4v-32h-8v28h-48v-48h28v-8h-32l0,0c-2.2,0-4,1.8-4,4v56C14.8,83.3,16.6,85.1,18.8,85.1z"></path> <polygon fill="currentColor" points="45.7,48.7 51.3,54.3 77.2,28.5 77.2,37.2 85.2,37.2 85.2,14.9 62.8,14.9 62.8,22.9 71.5,22.9"></polygon></svg></div> <div class="last-updated"><span class="prefix">Last Updated:</span> <span class="time">2/1/2020, 7:10:15 AM</span></div></footer> <div class="page-nav"><p class="inner"><span class="prev">
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